# Analysis and Geometry in Metric Spaces: Sobolev Mappings, the Heisenberg Group, and the Whitney Extension Theorem

Zimmerman, Scott (2017) Analysis and Geometry in Metric Spaces: Sobolev Mappings, the Heisenberg Group, and the Whitney Extension Theorem. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

 Preview
PDF

## Abstract

This thesis focuses on analysis in and the geometry of the Heisenberg group as well as geometric properties of Sobolev mappings. It begins with a detailed introduction to the Heisenberg group. After, we see a new and elementary proof for the structure of geodesics in the sub-Riemannian Heisenberg group. We also prove that the Carnot-Carath\'{e}odory metric is real analytic away from the center of the group.

Next, we prove a version of the classical Whitney Extension Theorem for curves in the Heisenberg group. Given a real valued function defined on a compact set in Euclidean space, the classical Whitney Extension Theorem from 1934 gives necessary and sufficient conditions for the existence of a $C^k$ extension defined on the entire space. We prove a version of the Whitney Extension Theorem for $C^1$, horizontal curves in the Heisenberg group.

We then turn our attention to Sobolev mappings.
In particular, given a Lipschitz map from a compact subset $Z$ of Euclidean space into a Lipschitz connected metric space, we construct a Sobolev extension defined on any bounded domain containing $Z$.

Finally, we generalize a classical result of Dubovitski\u{\i} for smooth maps to the case of Sobolev mappings. In 1957, Duvovitski\u{\i} generalized Sard's classical theorem by establishing a bound on the Hausdorff dimension of the intersection of the critical set of a smooth map and almost every one of its level sets. We show that Dubovitski\u{\i}'s theorem
can be generalized to $W_{\rm loc}^{k,p}(\mathbb{R}^n,\mathbb{R}^m)$ mappings for all positive integers $k$ and $p>n$.

## Share

Citation/Export: Select format... Citation - Text Citation - HTML Endnote BibTex Dublin Core OpenURL MARC (ISO 2709) METS MODS EP3 XML Reference Manager Refer

## Details

Item Type: University of Pittsburgh ETD
Status: Unpublished
Creators/Authors:
Zimmerman, Scottsrz5@pitt.edusrz5
ETD Committee:
Committee ChairHajlasz, Piotrhajlasz@pitt.eduhajlasz
Committee MemberTyson, Jeremytyson@illinois.edu
Committee MemberLennard, Christopherlennard@pitt.edulennard
Committee MemberDeBlois, Jasonjdeblois@pitt.edujdeblois
Date: 2 July 2017
Date Type: Publication
Defense Date: 3 April 2017
Approval Date: 2 July 2017
Submission Date: 30 March 2017
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 141
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: analysis, sub-Riemannian geometry, metric space, Sobolev, extension, Heisenberg group, Whitney extension theorem
Date Deposited: 02 Jul 2017 21:07
URI: http://d-scholarship.pitt.edu/id/eprint/31343

### Available Versions of this Item

• Analysis and Geometry in Metric Spaces: Sobolev Mappings, the Heisenberg Group, and the Whitney Extension Theorem. (deposited 02 Jul 2017 21:07) [Currently Displayed]